Controllability and Observability of a Well-Posed System Coupled With a Finite-Dimensional System

We consider coupled systems consisting of a well-posed and strictly proper (hence regular) subsystem and a finite-dimensional subsystem connected in feedback. The external world interacts with the coupled system via the finite-dimensional part, which receives the external input and sends out the output. Under several assumptions, we derive well-posedness, regularity, exact (or approximate) controllability and exact (or approximate) observability results for such coupled systems.